Conic Section Equations

Why This Matters

Conic sections aren't just abstract curves. They're the mathematical foundation for satellite dish design, planetary orbits, and headlight reflectors. In Honors Algebra II, you need to recognize these curves from their equations, convert between forms, and understand how changing parameters affects the shape. The core concepts here (distance relationships, symmetry, and the interplay between algebraic and geometric representations) will follow you straight into precalculus and calculus.

Don't just memorize formulas. For each conic, know what geometric property defines it (equidistant points? constant sum of distances?) and how to identify it from an equation. When you see a problem, ask yourself: What's the center? What's the orientation? What do the constants tell me about the shape?

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Curves Defined by Equal Distance

These conics are built on the simplest geometric idea: points that maintain a specific distance relationship to a center or axis.

Circle Equation

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$ — every point is exactly $r$ units from the center $(h, k)$
• Radius $r$ determines size. The equation uses $r^2$, so don't forget to take the square root when finding the actual radius. If you see $= 49$, the radius is 7, not 49.
• Only conic with equal coefficients — when $A = C$ and $B = 0$ in general form, you've got a circle

Parabola Equation (Vertical Axis)

• Standard form: $(x - h)^2 = 4p(y - k)$ — the set of points equidistant from the focus and directrix
• Parameter $p$ is the directed distance from vertex to focus. Positive $p$ opens upward, negative opens downward.
• Vertex at $(h, k)$ with focus at $(h, k + p)$ and directrix at $y = k - p$

Parabola Equation (Horizontal Axis)

• Standard form: $(y - k)^2 = 4p(x - h)$ — same distance property, but oriented left-right
• Sign of $p$ determines direction — positive opens right, negative opens left
• Focus at $(h + p, k)$ with directrix at $x = h - p$. The squared variable tells you the axis of symmetry.

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Curves Defined by Sum or Difference of Distances

Ellipses and hyperbolas are defined by their relationship to two foci. The key is whether you're adding or subtracting those distances.

Ellipse Equation

• Standard form: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ — the sum of distances from any point on the curve to both foci is constant (equals $2a$, where $a$ is the semi-major axis)
• Larger denominator determines orientation. If $a^2$ is under the $x$-term and $a^2 > b^2$, the major axis is horizontal. If the larger denominator is under the $y$-term, the major axis is vertical. The variable $a$ always refers to the semi-major axis, regardless of which fraction it sits beneath.
• Relationship: $c^2 = a^2 - b^2$ gives the focal distance, where $c$ is the distance from center to each focus

Hyperbola Equation (Horizontal Transverse Axis)

• Standard form: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ — the absolute difference of distances to the foci is constant (equals $2a$)
• The positive term determines orientation. The $x$-term is positive here, so the branches open left and right.
• Asymptotes: $y - k = \pm\frac{b}{a}(x - h)$ — these lines guide the hyperbola's shape as it extends outward

Hyperbola Equation (Vertical Transverse Axis)

• Standard form: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ — same difference property, but branches open up and down
• Relationship: $c^2 = a^2 + b^2$ for hyperbolas. Note this is addition, unlike ellipses.
• Asymptotes: $y - k = \pm\frac{a}{b}(x - h)$ — the slope formula flips compared to horizontal hyperbolas because $a$ is now associated with the $y$-direction

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Classification and Conversion Tools

These formulas help you identify conic types and convert between equation forms.

General Form of a Conic Section

• Form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ — the universal equation that contains all conics
• Discriminant $B^2 - 4AC$ classifies the conic: negative = ellipse or circle, zero = parabola, positive = hyperbola

Converting general form to standard form requires completing the square. Here's the process:

1. Group all $x$-terms together and all $y$-terms together. Move the constant to the other side.
2. Factor out the leading coefficient from each group (e.g., factor 4 from $4x^2 - 16x$).
3. Complete the square inside each group. Remember to add the same value to both sides (accounting for the factored coefficient).
4. Rewrite each group as a squared binomial and simplify the right side.
5. If needed, divide both sides so the equation equals 1 (for ellipses and hyperbolas) or isolate the unsquared variable (for parabolas).

Eccentricity Formula

• Formula: $e = \frac{c}{a}$ — measures how "stretched" a conic is from circular
• Classification by eccentricity: $e = 0$ (circle), $0 < e < 1$ (ellipse), $e = 1$ (parabola), $e > 1$ (hyperbola)
• Higher eccentricity = more elongated. Earth's orbit has $e \approx 0.017$, which is nearly circular. Halley's Comet, by contrast, has $e \approx 0.967$, producing a very elongated ellipse.

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Focus-Directrix Relationships

The directrix provides an alternative way to define parabolas using the equal-distance property.

Directrix for Vertical Parabola

• Equation: $y = k - p$ — a horizontal line located $|p|$ units from the vertex, on the opposite side from the focus
• Defines the parabola geometrically: every point on the curve is equidistant from the focus $(h, k + p)$ and this directrix
• The directrix is always perpendicular to the axis of symmetry.

Directrix for Horizontal Parabola

• Equation: $x = h - p$ — a vertical line located $|p|$ units from the vertex, on the opposite side from the focus
• Same distance property applies: point-to-focus distance equals point-to-directrix distance for every point on the parabola
• Sign of $p$ matters. If $p > 0$, the focus is right of the vertex and the directrix is left. Flip for $p < 0$.

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Quick Reference Table

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Self-Check Questions

1. Given the equation $\frac{(x-2)^2}{16} + \frac{(y+3)^2}{25} = 1$, is the major axis horizontal or vertical? What is the center?

2. How can you tell the difference between an ellipse and a hyperbola just by looking at the standard form equation?

3. Compare the formulas for $c$ in ellipses versus hyperbolas. Why does this difference make geometric sense?

4. A parabola has vertex at $(1, -2)$ and focus at $(1, 0)$. Write its equation and find the directrix.

5. You're given $4x^2 + 9y^2 - 16x + 18y - 11 = 0$. What type of conic is this, and what's your first step to convert it to standard form?